Conic Theta Functions and their relations to classical theta functions

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1 Conic Theta Functions and their relations to classical theta functions International Number Theory Conference in Memory of Alf Van der Poorten The University of Newcastle, Australia Sinai Robins Nanyang Technological University March 16, 2012 Joint work with A. Folsom and W. Kohnen

2 Def. A polyhedral cone K is the non-negative real span of a finite number of vectors in R d. Cone K θ

3 Extending angles to higher dimensions: Defining the solid angle at a vertex of a cone sphere centered at the origin This is a geodesic t r i a n g le o n t h e sphere, representing the solid angle ω K Cone K

4 Def. The solid angle for the cone K is ω K := vol(s K). vol(s) Thus 0 ω K 1.

5 Def. The solid angle for the cone K is ω K := vol(s K). vol(s) Thus 0 ω K 1. Equivalently, ω K is the normalized volume of a spherical polytope.

6 Def. The solid angle for the cone K is ω K := vol(s K). vol(s) Thus 0 ω K 1. Equivalently, ω K is the normalized volume of a spherical polytope. A third equivalent condition is that ω K = K eπ x 2 dx.

7 Motivation Problem 1. Which polyhedral cones K give rise to spherical polytopes whose volume is a rational number?

8 Motivation Problem 1. Which polyhedral cones K give rise to spherical polytopes whose volume is a rational number? Problem 2. Considering a certain conic theta function Φ K, associated to any polyhedral cone K, how close is Φ K to a modular form?

9 Let L be a rank d lattice in R d,letk R d be a polyhedral cone, and let τ H, the upper complex half plane.

10 Let L be a rank d lattice in R d,letk R d be a polyhedral cone, and let τ H, the upper complex half plane. We define the conic theta function Φ K,L (τ) = m K L e πiτ m 2.

11 Conic theta functions are discretized volumes of spherical polytopes

12 Conic theta functions are discretized volumes of spherical polytopes Lemma. (Folsom, Kohnen, R. 2011) Let L R d be a full rank lattice. Then lim t d 2 ΦK,L (it) = t 0 ω K detl.

13 Conic theta functions are discretized volumes of spherical polytopes Lemma. (Folsom, Kohnen, R. 2011) Let L R d be a full rank lattice. Then lim t d 2 ΦK,L (it) = t 0 ω K detl. Proof. Look carefully at the Riemann sum definition of ω K := K e π x 2 dx.

14 By contrast, the classical theta function associated to the lattice L R d is defined by: Θ L (τ) = m L e πiτ m 2.

15 By contrast, the classical theta function associated to the lattice L R d is defined by: Θ L (τ) = m L e πiτ m 2. Fact. If L is an even, integral lattice, then Θ L (τ) is a modular form of weight d/2 on Γ 0 (N).

16 We define R to be the ring of all finite, rational linear combinations of theta functions Θ L, varying over all d-dimensional even integral lattices L R d.

17 We define R to be the ring of all finite, rational linear combinations of theta functions Θ L, varying over all d-dimensional even integral lattices L R d. Theorem. (A.Folsom, W.Kohnen, R. 2011) If K is the Weyl chamber of a finite reflection group, then the conic theta function Φ K,2L (τ) lies in the graded ring R.

18 On the other hand, if the solid angle ω K (a.k.a. the volume of a spherical polytope) if irrational, then it turns out that Φ K,L (τ) is not a modular form.

19 More precisely, let K R d be a polyhedral cone, and let L := A(Z d ) be an even integral lattice of full rank. Theorem. (A. Folsom, W. Kohnen, R. 2011) If ω K deta is irrational, then Φ K,L is not a modular form of weight k on any congruence subgroup, and any k 1 2 Z, k 1 2.

20 Tools 1. The q-expansion principle, due to Deligne and Rapoport, tells us that if an integer weight modular form f has rational Fourier coefficients at the cusp i, then the Fourier expansion of f at all other cusps must also have rational coefficients. 2. Combinatorial geometry of cones.

21 1. If ω K deta Open problems Q, we do not have proofs of non-modularity for Φ K,L, except for some special cases. (recalling that L := A(Z d )) 2. If we have an integer cone K, does its conic theta function Φ K,L lie in the ring R of rational linear combinations of classical theta functions?

22 Thank you